Connection between the Bellman equation for the action value function $q_\pi(s,a)$ and expressing $q_\pi(s,a) = q_\pi(s, a,v_\pi(s'))$
Your understanding of the Bellman equation is not quite right. The state-action value function is defined as the expected (discounted) returns when taking action $a$ in state $s$. Now, in your equation (2) you have conditioned on taking action $a'$ in the inner expectation - this is not what happens in the state-action value function, you do not condition on knowing $A_{t+1}$, it is chosen according to the policy $\pi$ as per the definition of a bellman equation.
If you want to see a 'recursion' between state action value functions, note that
$$v_\pi(s) = \sum_a \pi(a|s)q_\pi(s,a)\;,$$
Your equation (5) is incorrect -- you need to drop the outter sum over $a$ as we have conditioned on knowing $a$. I will drop the $\pi$ subscripts for ease on notation, and we can see a 'recursion' for state-action value functions as:
$$q(s,a) = \sum_{s',r}p(s',r|s,a)\left(r + \gamma \left[\sum_{a'} \pi(a'|s')q(s',a')\right]\right)\;.$$